Is watching a video that contains something forbidden, while overlooking the forbidden and benefiting from the permissible, and contributing to an increase in the number of views, considered participation in the sin of subsequent viewers or the video's owner?

If the music in the videos is confined to specific, avoidable sections, then there is no harm in benefiting from them. There appears to be no reason to deem you sinful merely for accessing those sites. Increasing the video's popularity is permissible as long as it can be utilized in a lawful manner. You can advise the website administration to separate the music from the videos.